Mackie Mayor is a great shout - you might also find the Art Gallery and MOSI good options as they have good kids activities in general so might have some extra goodies for little ones in the cafes, also Whitworth Gallery if you go a little bit out of the centre. If you wrap up then Mayfield Park could be a good option too.
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Just outside town is the Lowry, it has a cafe and play area upstairs
The mackie mayor has a little play bit under the stairs that my little girl likes to play in. And the coffee is good there. Other than that I can’t think of anywhere with a play bit for kids.